critical metastability and destruction of the splitting in ... · explicitly compute the wave...

Journal of Statistical Physics, Vol. 103, Nos. 1�2, 2001

Critical Metastability and Destruction of the Splittingin Non-Autonomous Systems

Vincenzo Grecchi1 and Andrea Sacchetti2

Received January 7, 2000; final August 1, 2000

We study a periodically driven double well model. As in the case of autonomousmodels, previously treated in a joint paper with A. Martinez, (7) we have thedestruction of the splitting for critical metastability. The relevance of the modelfor the understanding of the red shift in the inversion line of the molecule ofammonia is shortly discussed. We show that, in order to have a reasonablebehavior of the metastability as a function of the frequency, a non-monochromaticperturbation is needed.

KEY WORDS: Resonances; localization; double well; time dependentHamiltonian; periodic external fields; quantum stability.

1. INTRODUCTION

In this paper we study the splitting destruction in a symmetric double wellmodel subjected to an external time dependent perturbation. The physicalmotivation is the problem of the localization of symmetric moleculesinduced by collisions, as observed in the ammonia molecule NH3 , with theassociated ``red shift effect,'' i.e., the gradual vanishing of the splitting forincreasing pressure (see ref. 8, see also the recent review paper on thisproblem by Wightman(15)). Although our model is not completely physical,in fact, we assume that the perturbation is periodic and we don't take intoaccount non-linear effects, it can be useful in order to understand if thephenomenon of the vanishing of the splitting is already present in the simplest

339

0022-4715�01�0400-0339�19.50�0 � 2001 Plenum Publishing Corporation

1 Dipartimento di Matematica, Universita� di Bologna, Piazza di Porta S. Donato 5, I-40127Bologna, Italy; e-mail: Grecchi�dm.unibo.it

2 Dipartimento di Matematica, Universita� di Modena e Reggio Emilia, Via Campi 213�B,I-41100 Modena, Italy; e-mail: Sacchetti�unimo.it

cases. We underline that our research is not in disagreement with the notionof de-coherence (see ref. 6 for a review), very useful for understanding theclassical behavior of certain microscopic systems. Indeed, since our non-autonomous model yields localization of the states, as in the classical case,it can be considered as an explicit model for de-coherence.

Here, we make use of new techniques for the analysis of non-autono-mous Hamiltonian systems recently developed by Soffer and Weienstein.(14)

By assuming that the state is initially prepared on the two ground states ofthe double well, we compute the solution of the time-dependent Schro� dingerequation with the rigorous control of the error (see Theorem 2 in Section 3).In particular, we obtain that the time behavior of the wave function, fortimes of the order of the unperturbed beating period, is described by meansof two complex eigenvalues of a matrix independent of time. The imaginarypart of the eigenvalues is related to the metastability of the state. Splittingvanishing occurs when these eigenvalues coincide.

As appears in an explicit model (see Sections 5 and 6), for a fixedstrength of the perturbation, of the order of the square root of the unper-turbed splitting, we have increasing metastability for increasing frequencyof the periodic perturbation in a certain range of values (see Theorem 5 inSection 6). What it is peculiar to our research is the relevant role of themetastability and the existence of a critical value for the metastability inorder to have the vanishing of the splitting. The general rule we have foundcan be easily understood and can be stated in a simple way: the criticalvalue of the mean life is equal to the beating period of the unperturbed double well.

We underline that localization on one well, which is easy to obtainwith simpler static models, (4) appears here when the strength of the pertur-bation is much larger than the square root of the splitting; in contrast, forperturbations with strength smaller than the square root of the splitting(Theorem 3), we observe the unperturbed beating effect.

We remark that, in order to obtain the desired results, we don't makeuse of resonance effects and we don't need to assume that the external per-turbation is monochromatic. Thus, our research is completely differentfrom others (see refs. 3 and 10, see also ref. 9 and the references therein) onthe same subject of destruction of the splitting.

The paper is organized as follows. In Section 2 we state the principalassumptions on the potential and we introduce the notation. In Section 3we state our main results (Theorems 1 and 2). In Section 4 we give theproof of the theorems. In Section 5 we introduce a simple one-dimensionalmodel satisfying the technical assumptions of Section 2. In Section 6 weexplicitly compute the wave function for the model given in Section 5 andwe obtain the vanishing of the splitting (Theorem 5) for some values of thefrequency and of the strength of the perturbation.

340 Grecchi and Sacchetti

2. ASSUMPTIONS AND NOTATIONS

In this paper we consider the Schro� dinger equation

i�,4 =(H\+W ) ,, ,(t) # H, t # R (1)

where H is the Hilbert space L2(Rn, dx), n�1,

H\=&�2

2m2+V\

[H\]\ # I is a family of self-adjoint (time-independent) operators on thedomains D\/H, I�R+ and +� # I� where I� denotes the closure of I,and W is a time-dependent perturbation.

Hypothesis H1: Assumption on V. We assume that the spectrumof H\ is given by a _(H\)=[*\

1 , *\2] _ [0, +�) where *\

1, 2 are twonegative simple eigenvalues *\

1<*\2<0 such that

lim\ � +�, \ # I

*\1= lim

\ � +�, \ # I*\

2=*� <0 (2)

Remarks.

�� We denote |\= 12 (*\

2&*\1) and 0\= 1

2 (*\2+*\

1), from (2) it followsthat |\ � 0 and 0\ � *� as \ � +�, \ # I;

�� In general, the above assumption is satisfied when V\ is a symmetricdouble well potential such that:

lim|x| � �

V\(x)=0

In particular, for a suitable choice of the potential we have that H\ hasonly two eigenvalues and, in the limit of large barrier between the wells, wehave that |\

te&c\A where the Agmon distance \A between the wells goesto infinity (see, for instance, ref. 7 and the references therein).

Hypothesis H2: Assumptions on W. The time-dependent per-turbation W#W(t, x) has the form

W(t, x)='g(x)+=v(+t) f (x), =, '>0

where = and ' are small parameters and the frequency + is a parameterbelonging to a set M/(0, +�). We assume that:

341Critical Metastability and Destruction

(i) f (x) and g(x) are real-valued piece-wise continuous functionswith compact support contained in a compact set U;

(ii) v(t) is a real-valued periodic function, with period L, such thatc0=0 (i.e., the mean value of v(t) is zero) and cn=0 for any |n|>N, forsome positive integer N, where cn , n # Z, are the Fourier coefficients of v(t);

(iii) the frequencies + # M are non-resonant, that is there exists d>0such that

|n++0\|>d, \n=0, \1,..., \N, + # M and \ # I

In particular we have that |n++0\| # [d, D] for any n, + and \ andsome D>d.

Notation.

�� For the sake of simplicity, we make the choice of units such that�=1, 2m=1 and L=2?;

�� & }&H and ( } , } ) H denote the norm and the scalar product of theHilbert space H, (x) =- 1+|x|2;

�� /A(x) denotes the characteristic function on the set A, i.e.,/A(x)=1 if x # A and /A(x)=0 if x{A;

�� we drop the dependence on \, x, + and t when this does not causemisunderstanding, in particular we denote |\ and 0\ by | and 0;

�� we set ;=max(', =) and ;� =max(;, |), we denote by C a genericpositive constant independent of t, ', = and \ which need not have the samevalue throughout the paper;

�� we denote by Pc the projection operator on the eigenspaceassociated to the essential spectrum of H\ , _ess(H\)=[0, +�): i.e., Pc=1&�\

1(�\1 , } )&�\

2(�\2 , } ) where �\

1, 2 are the normalized eigenvectors ofH\ associated to *\

1, 2 ;

�� we formally denote

Km=[H\&(0+m++i0)]&1=w& lim! � 0+

[H\&(0+m++i!)]&1

v let M be a generic 2_2 matrix, with elements denoted by M\, \ ,and let A be a generic column matrix, with elements denoted by A\ , i.e.:

M=\M+, +

M&, +

M+, &

M&, &+ and A=\A+

A&+


WM X and WAX respectively denote WM X=max |M\, \ | and WAX=max |A\ |

�� let M(t) be a matrix-valued function defined for t�0, then wedenote

[M ](t)= sup0�{�t

WM({)X

�� let .: R+ � H be a vector-valued function defined for t�0 andsuch that (x) \_ .(t) # H, where _>0 is fixed, then we denote

(.)$t)= sup0�{�t

&(x) \_ .({)&H

By definition [M ](t) and (.)\(t) are monotone non-decreasing functions.We state now our main assumptions:

Hypothesis H3: Time-decay assumptions. There exist _>0and r>2 such that for any ,, such that (x) _ , # H, the followingestimates uniformly hold with respect to \ # I, * and t�0:

&(x) _ [H\&*]&1 Pc ,&H�C &(x) _ ,&H ,

\* # [&D, &d] (3)

&(x) &_ e&iH\ t Pc ,&H�C(t) &r+1 &(x) _ ,&H (4)

&(x)&_ e&iH\t[H\&(*+i0)]&1 Pc,&H�C(t) &r+1 &(x) _ ,&H (5)

for any * # [d, D]. Moreover, we assume also that for any ,1 and ,2 withcompact support then

|(,1 , [H\&*]&1 Pc,2) H |�C, \* # [&D, &d] _ [d, D] (6)

Remarks.

�� In fact, in order to prove our main result stated below it is suf-ficient to assume the weaker condition that (4) and (5) uniformly hold forany t # [0, T ] where T=T ($=2?�|.

�� For any fixed \, condition (4) is true for any n�7 provided that|V\(x)|�C(x) &s for some s>0 (see Theorem 2.1 in ref. 12)), and it isgenerically true for any n (see ref. 11 and Theorem 7.6 in ref. 12) providedthat *=0 is neither an eigenvalue nor a resonance of H\ (property (5) canbe proved as a consequence of (4) as done in Appendix A in ref. 13). Wediscuss in Section 5 the validity of the time-decay assumptions uniformlywith respect to \ for an explicit model.


3. MAIN RESULTS

Let �1, 2 # H be the normalized eigenvectors of H associated to *1, 2

(let us drop the dependence on \); let

�\=1

- 2(�1\�2)

be the single-well ground states, they are such that

(�\ , �\) H=1, (�\ , ��) H=0 and H�\=0�\&|��

The solution ,(t) # H of the time-dependent Schro� dinger equation (1) canbe written as

,(t)=a+(t) �++a&(t) �&+,c(t) (7)

where ,c=Pc,, that is

(,c(t), �\) H=0, \t # R

We assume that the state is initially prepared on the two single-well groundstates:

Hypothesis H4. The initial state ,0=,(0) is such that ,0c=Pc,0=0.

By substituting , by (7) in Eq. (1) and projecting the resulting equa-tion on �\ and on the eigenspace associated to the essential spectrum, weobtain the following system of equations for a\ and ,c :

ia* +=0a+&|a&+a+(�+ , W�+) H+a&(�+ , W�&) H

+(�+ , W,c) H{ ia* &= &|a++0a&+a+(�& , W�+) H

+a&(�& , W�&) H+(�& , W,c) H

i,4 c=H\,c+a+Pc W�++a& PcW�&+PcW,c

satisfying to the initial conditions

a0\=(�\ , ,0) H , ,0

c=0

Let A$t)=a\(t) ei0t, A be the column matrix with elements A\ ,_1=( 0

110) be the first Pauli matrix, F and G be the 2_2 symmetric

matrices with elements

F\, \=(�\ , f�$ H and G\, \=(�\ , g�\) H


Rc be the column matrix with elements Rc, $t)=ei0t(�\ , W,c) H ; thenthe above system can be written in the form

{iA4 =&|_1 A+'GA+=v(+t) FA+Rc

i,4 c=H\,c+a+PcW�++a&PcW�&+PcW,c(8)

It is a matter of integration by parts and use of the second differentialequation of the system (8), to obtain the following preliminary result.

Theorem 1. Let ;=max(=, ') and

4m\, \(+)=( f�\ , KmPc f�$ H , m{0 (9)

40\, \=(g�\ , K0Pcg�\) H (10)

Let

,(t)=A+(t) e&i0t�++A&(t) e&i0t�&+,c(t)

be the solution of Eq. (1) where ,c(t)=Pc,(t). Then A=( A+A&

) is thesolution of the equation

A4 =&iMA+iMper(+t) A+R (11)

where Mper(t) is a periodic 2_2 matrix, with period 2?, with mean valuezero and such that WMper(t)X�C; for some positive constant C independentof \ and t; M is 2_2 matrix independent of t given by

M=&|_1+'G+=2U+'2S (12)

where the elements of U and S are given by

U\, \=& :N

m{0, m=&N

|cm |2 4m\, \(+), S\, \=&40

\, \

R=R(A4 , A, t) is a remainder term.We consider, for the present, the linear differential equation with

periodic coefficient

B4 =[&iM+iMper(+t)] B, B(0)=A(0) (13)

It is well know (see Theorem 5.1, Chap. 3, ref. 5) that the solution of thisequation has the form B(t)=P(+t) e&iNtA(0) for some constant matrix Nand periodic matrix P. We assume that


Hypothesis H5. Let n1, 2 be the eigenvalues of the matrix N andlet #=max[In1 , In2], we assume, in the limit of small perturbation andlarge \, that #�C| and |P(t)|�C, for any t, for some C>0 independentof \, = and '.

We state now our main result:

Theorem 2. Let + # M be fixed, let ;=max(', =) and ;� =max(|, ;). From the above Hypotheses H1, H2, H3, H4, and H5, in the limitof small perturbation and large \, \ # I, such that ;3�| � 0, then it followsthat the solution of (11) is given by

A(t)=B(t)+RA(t) (14)

where B(t) is the solution of Eq. (13) and where RA(t) is a remainder termsatisfying to the following uniform estimate

WRA(t)X�C WA(0)X ;� ;2�|, \t # [0, T ], T=2?|

(15)

for some positive constant C independent of =, ' and \. Moreover, itfollows also the estimate (,c)&(t)�C; WA(0)X for any t # [0, T ].

4. PROOF OF THE THEOREMS

The proof of the theorems follows the line of the paper, (14) adaptedhere to our model. In particular, we obtain the estimate (15) uniformlywith respect to \.

Proof of Theorem 1. From the second equation of (8) we can write

,c(t)=&i[,0(t)+,+(t)+,&(t)+,d (t)

where

,0(t)=eiHt,c(0)#0

,d (t)=|t

0e&H(t&s)PcW(s, } ) ,c(s) ds (16)

,\(t)=|t

0e&iH(t&s)PcW(s, } ) a\(s) �\ ds (17)


If we set hn(x)= f (x), #n==cn , n{0, and h0(x)= g(x), #0=', then we canwrite

W(t, x)= :N

n=&N

#n ein+thn(x) (18)

From this fact and assuming, for the present, that 0+n+ � _ess(H ), for anyn=0, \1,..., \N, then it follows that

,\(t)= :N

n=&N

#&n |t

0e&i[H(t&s)+(0+n+) s]Pch&n A\(s) �\ ds

= :N

n=&N

i#&n {&e&i(0+n+) tKnPch&n�\A\(t)

+e&iHtKnPch&n�\A\(0)

+|t

0e&i[H(t&s)+(0+n+) s]KnPch&n A4 \(s) �\ ds= (19)

by integrating by parts. If n is such that 0+n+ # _ess(H ), the aboveformula is still true; indeed, it follows in the same way by taking n++i!,!>0, and the limit ! � 0+. Let

Rc, \=ei0t(�\ , W,c) H=:\, ++:\, &+:d, \ (20)

where :\, +=&iei0t(�\ , W,+)H , :\, &=&ie i0t(�\ , W,&) H and:d, \=&iei0t(�\ , W,d) H . From this and from (17) we have that

:\, += :N

m=&N

&iei(0+m+) t#m(�\ , hm,+) H

=:0\, +A+(t)+:1

\, + A+(t)+:2\, +A+(0)+:3

\, +

where

:0\, +=& :

N

m=&N

#m#&m(hm �\ , KmPch&m�+) H

:1\, +=& :

N

n, m=&N, m{n

#m#&ne&i(n&m) +t(hm�\ , Kn Pch&n�+) H

:2\, += :

N

m, n=&N

#m#&nei(0+m+) t(hm�\ , e&iHtKnPc h&n�+) H

:3\, += :

N

m, n=&N

#m#&nei(0+m+) t

_|t

0(hm�\ , e&i[H(t&s)+(0+m+) s]Kn Pch&n �+) H A4 +(s) ds


and a similar result for :\, & follows. From this equation then (11) followswhere the time independent matrix M is obtained by collecting the terms|_1 , 'G and :0

\, \ , the periodic matrix Mper(t) is obtained by collectingthe terms v(+t) F and :1

\, \ , the remainder term R is obtained by collectingthe terms :2

\, \ , :3\, \ and :d, \ :

R\=:2\, +A+(0)+:2

\, &A&(0)+:3\, ++:3

\, &+:d, \ (21)

We conclude the proof of Theorem 1 by underlining that Mper(t) is periodi-cally dependent on t, with mean value 0 and it is such that WMper(t)X�C;for some C>0 independent of =, ' and \ because of (6).

Proof of Theorem 2. Solution of (11), satisfying to the initial con-dition A(0), is given by (14) where the term RA(t) is given by (seeTheorem 3.1, Chap. 3, ref. 5)

RA(t)=P(t) |t

0e&N(t&{)P&1({) R({) d{

In order to obtain a bound of the remainder term RA we give a sequenceof technical Lemmas.

Lemma 1. There exists a positive constant C, independent of t,such that

WRA(t)X�Ct[R](t) and [RA](t)�Ct[R](t), \t # [0, T ] (22)

Proof. Let #=max[In1 , In2], where n1, 2 are the two eigenvaluesof N. From the definition of RA , from the facts that #�C| and WP(t)X�Cfor any t, then it follows that

WRA(t)X�C |t

0e#(t&{) WR({)X d{�Ct[R](t)

proving the first inequality of (22). The second one immediately follows asa result of the first one, indeed

[RA](t)= sup{ # [0, t]

WRA({)X� sup{ # [0, t]

C{[R]({)�Ct[R](t)

Lemma 2. For any ,: R � H, such that (x) _ ,(t) # H, and |*| #[d, D] there exists a positive constant C independent of t and \ such that

|t

0&(x) &_ e&iH(t&s)Pc,(s)&H ds�C(,)+(t)


and

|t

0&(x) &_ e&iH(t&s)[H&(*+i0)]&1 Pc,(s)&H ds�C(,)+(t)

Proof. From (4) it follows that

|t

0&(x) &_ e&iH(t&s)Pc,(s)&H ds�C |

t

0(t&s) &r+1 ds(,)+(t)

�C(,)+(t)

for any t since r>2. In the same way the second estimate follows from (5),when * # [d, D], and from (3) and (4) when * # [&D, &d].

Lemma 3. For any t>0 there exists a positive constant C independentof t, ; and \ such that

(,c)&(t)�C;[A](t) (23)

Proof. Given the definition of ,c , given formulas (16) and (17), giventhe Schwartz inequality and given the first estimate of Lemma 2, it followsthat

&(x)&_ ,c({)&H�&(x) &_ ,+({)&H+&(x) &_ ,&({)&H

+&(x) &_ ,d ({)&H

�C[A]({)[ sup0�s�{

&(x) _ W(s, x) �+&H

+ sup0�s�{

&(x)_ W(s, x) �&&H]

+C sup0�s�{

&(x) _ W(s, x) ,c&H

�C;[A]({)+C sup0�s�{

&(x) _ W(s, x) ,c&H

=C;[(,c)&({)+[A]({)]

since

sup0�s�{

&(x) _ W(s, x) �\&H�C; sup0�s�{

&(x) _ /U(x) �\&H�C;


and

sup0�s�{

&(x) _ W(s, x) ,c&H� sup0�s�{

&(x)_ W(s, x)(x) _&H } &(x) &_ ,c&H

�;C sup0�s�{

&(x) &_ ,c&H=;C(,c)&({)

because W(t, x) has compact support contained in U for any t. If weremark that (,c)&(t) and [A](t) are monotone non-decreasing functionsthen (23) follows.

Lemma 4. For ; small enough it follows that

[R](t)�C;2((t) &r+1 WA(0)X+;� [A](t)), where ;� =max(|, ;)

(24)

for some positive constant C independent of ;, \ and t.

Proof. In order to prove (24) we separately estimate the terms :2\, \ ,

:3\, \ and :d, \ defined in the proof of Theorem 1. In order to consider the

term :2\, \ we set .n, \=KnPch&n�\ . Let, for the present, be n such that

n++0�&d, from (3) it follows that (x)_ .n, \ # H since (x) _ hn�\ # H;from this fact and from (4) it follows that

|((x) _ hm�\ , (x) &_ e&iHtPcKnPch&n�\) H |

�&(x) _ hm�\&H &(x) &_ e&iHtPc.n, \&H

�C &(x) _ hm�\&H &(x) _ KnPch&n �\&H (t) &r+1

�C(t) &r+1

The same result directly follows from (5) in the case 0+n+�d. Therefore,we can conclude that

|:2\, \ |�C;2(t) &r+1 (25)

For what regards the term :3\, \ we remark that from (11) it follows that

WA4 (s)X�C;� WA(s)X+WR(s)X , where ;� =max(|, !)

then, applying the result of Lemma 2 to the vector e&i(0+n+) sA4 $s) h&n�\ ,we obtain that

|:3\, \ |�C;2 :

N

n=&N

(A4 \(s) h&n�$+(t)�C;2(;� [A](t)+[R](t)) (26)


For what concerns the last term :d, \ we apply Lemma 2 to ,=W,c

obtaining

|:d, \ |= }�(x) _ W�\ , (x) &_ |t

0e&iH(t&s)PcW(s, } ) ,c ds�H }

�C; |t

0&(x) &_ e&iH(t&s)PcW(s, } ) ,c &H ds

�C;2(/U ,c)+(t)�C;2(,c)&(t)

since &(x)_ /U ,c&H�&(x) _ /U(x) _&H } &(x) &_ ,c&H . From this andfrom Lemma 3 we finally obtain

|:d, \ |�C;3[A](t) (27)

Collecting the results (25), (26) and (27) and relation (21) we obtain

WR(t)X�;2C[(t) &r+1 WA(0)X+[R](t)+;� [A](t)]

from which Lemma 4 follows for ; small enough.

Lemma 5. In the limit of small perturbation and large \, such that;3�|<<1, then there exists a positive constant C independent of \, = and' such that

[A](t)�C WA(0)X, \t # [0, T ], T=2?|

(28)

Proof. From Eqs. (14), (22) and (24) and recalling that #�C| thenit follows that for any 0�t�T=2?�|

[A](t)�C[ sup0�{�t

e#{ WA(0)X+[RA](t)]

�C[WA(0)X+t[R](t)]

�C[WA(0)X+;2t((t) &r+1 WA(0)X+;� [A](t))]

�C[WA(0)X(1+;2|r&2)+;� ;2|&1[A](t)]

for some positive constant C. From this the result follows since r>2 and;� ;2|&1=max(;2, ;3|&1)<<1.


Collecting all these results we are able to prove Theorem 2; indeed,from Eqs. (22), (24) and (28) it follows that

WRA(t)X�Ct[R](t)

�C;2((t) &r+2 WA(0)X+;� t[A](t))

�C;� ;2|&1 WA(0)X , \t # [0, T ]

Finally, the bound of (,c)& it follows as a direct result of Lemmas 3 and 5.

5. THE EXPLICIT MODEL

In this section we consider the following explicit one-dimensionalmodel: the double well potential is given by means of two attractive deltafunctions at x=\a with negative strength &b and one repulsive deltafunction at x=0 with positive strength \:

V\=&b $(x&a)&b $(x+a)+\ $(x)

where a>0 and b>0 are fixed and \>0 is large enough. We recall thatthe limit case \=+� corresponds to the case of two attractive $ functionsat x=\a with Dirichlet condition at x=0; that is H�=H +

D �H &D where

H \D is the Schro� dinger operator on the half line R\ with Dirichlet condi-

tion at x=0.(2) We prove now that this model satisfies to the assumptionsH1 and H3 of Section 2, and in particular the uniform bounds (3)�(6),provided that ab>1 and \ # I=[\~ , +�) with \~ >0 large enough. Let

y1=a, y2=0, y3=a, Gk(x)=i

2keik |x|, Ik>0

&\&1b

+i

2k+ &i

2keika &

i2k

ei2ka

1\, k=\ &i

2ke ika &\1

\+

i2k+ &

i2k

eika + (29)

&i

2kei2ka &

i2k

eika &\&1b

+i

2k+K\(x, y; k)=

i2k

eik |x& y|&1

4k2 :3

r, s=1

(1&1\, k)rs e ik |x& yr|eik | y& ys|


we recall that the resolvent operator [H\&z]&1, where H\=&d 2�dx2

+V\ and z=k2, is an integral operator with kernel K\ (see ref. 2):

([H\&k2]&1 ,)(x)=|R

K$x, y; k) ,( y) dy (30)

In order to study the spectrum of H\ we introduce the functions f(w), suchthat f(0)=1 and w f(w)=ew&1 for w{0, h(w)=1�ab&f(w) and g\(w)=h(w)&1�a\ [w f(w)+2+w�ab]. By means of a simple computation itfollows that

det 1\, k=a3

wh(w) g\(w), where w=i2ka

and that the elements of the inverse matrix 1&1\, k are given by

(1&1\, k)1, 1=(1&1

\, k)3, 3=&f(w)+1�a(1�b&1�$&w�a2\b

ag$w) h(w)

(1&1\, k)1, 2=(1&1

\, k)2, 1=(1&1\, k)2, 3=(1&1

\, k)3, 2=&ew�2

ag\(w)(31)

(1&1\, k)2, 2=

2+w�ab+w f(w)ag\(w)

(1&1\, k)1, 3=(1&1

\, k)3, 1=ew

a2\g\(w) h(w)

The discrete spectrum of H\ is given by the negative eigenvalues*=&w2�4a2 where w are the real and negative solutions of the equationsh(w)=0 and g\(w)=0. If ab>1 and \ is large enough then the equationh(w)=0 has only one real and negative solution w1 independent of \ andthe equation g\(w)=0 has only one real and negative solution w\

2 suchthat w\

2&w1=O(\&1). Therefore hypothesis H1 follows.We underline that when ab>1 then there exist a positive constant C

and \~ >0 such that

|h(0)|>C and |g\(0)|>C, \\ # I=[\~ , +�) (32)

in particular *=0 is neither an eigenvalue nor a resonance; moreover_ess(H$=_ac(H\)=[0, +�) (see ref. 2 again). We remark also that fromthe above formulas (31) it follows that 1&1

� &1&1\ =O(\&1) and

&[H�&k2]&1&[H\&k2]&1&�Ck \&1, Ik>0


therefore H\ converges to H� as \ � +� in norm resolvent sense. Hence,conditions (3) and (6) are uniformly true for any \ large enough.

In order to prove the bounds (4) and (5) uniformly with respect to \we observe that a direct computation gives that

:3

i=1

(1&1\, k)j, i=wF\

j (w), j=1, 2, 3 (33)

where we set

F\1(w)=F\

3(w)=a\f(w) f(w�2)&2�ab&\�b f(w�2)+2f(w)

2a2\g$w) h(w)(34)

and

F\2(w)=

a\h(w)[&f(w�2)+1�ab+f(w)]a2\g\(w) h(w)

(35)

Let # be an anti-clockwise curve, surrounding the absolute continuousspectrum _ac(H$=[0, +�), with endpoints +�+i0 and +�&i0; thespectral theorem gives that

(e&itH\Pc,)(x)=|#

e&izt([H\&z]&1 ,)(x) dz

=i |R+i0

2ke&ik2t dk |R

K\(x, y; k) ,( y) dy

=|R

U t\(x, y) ,( y) dy

where

Ut\(x, y)=i |

R+i02ke&ik 2tK\(x, y; k) dk

We prove that:

Lemma 6. Let a, b>0 such that ab>1 and let \~ >0 be largeenough in order to have (32). Then, the following asymptotic behavioruniformly holds for any \�\~ :

Ut\(x, y)=O([ut&1�2]3) where u=max((x) , ( y) ) (36)


Proof. In order to prove the above proposition we set ur, s=|x& yr |+| y& ys | and we remark that the Fourier transform of e&ik2t is- ? eiu2�4t�- it , then

U t\(x, y)=i |

R+i0e&ik 2teik |x& y| dk+V t

\(x, y)

=i - ?

- itei |x& y| 2�4t+V t

\(x, y)

V t\(x, y)=

12

:3

r, s=1|R+i0

e&ik2teik( |x& yr| +| y& ys| )(1&1\, k)r, s

dkk

=12

:3

r, s=1|R+i0

e&ik2teikur, s(1&1\, k)r, s

dkk

By means of the McLaurin series (1&1\, k)r, s=(1&1

\, 0)r, s+k(1&1\, 0)$r, s+

(1&1\, k)R

r, s , where (1&1\, k)R

r, s has a double (or higher) zero at k=0, we havethat Vt

\=V t0, \+V t

1, \+V tR, \ where

V t0, \(x, y)=

12

:3

r, s=1

(1&1\, 0)r, s |

R+i0e&ik2teikur, s

dkk

V t1, \(x, y)=

12

:3

r, s=1

(1&1\, 0)$r, s |

R+i0e&ik2teikur, s dk

V tR, \(x, y)=

12

:3

r, s=1|R+i0

(1&1\, k)R

r, s e&ik2teikur, sdkk

here $ denotes the derivative with respect to k. For what concerns thecomputation of the first term V t

0, \ we remark that the Cauchy theorem(where we perform the change of the path of integration k � e&i?�4k) gives:

|R+i0

e&ik2te iku dkk

=|R+i0

e&k2tee i?�4ku dkk

=&i?eiu2�4tW(ei?�4u�2 - t )


where W({)=(i�?) �R e&s2(ds�({&s)), I{>0 (see formula (7.1.4), ref. 1) is

such that (see formula (7.1.8), ref. 1)

H tr, s(x, y)=e iu2

r, s �4tW(ei?�4ur, s�2 - t )&1=_1+iu2

r, s

4t+O(u4�t&2)&

__1+iei?�4ur, s

2 - t 1 (3�2)+

(iei?�4ur, s)2

(2 - t )2 1 (2)+O([u�- t ]3)&&1

=iei?�4ur, s

2 - t 1 (3�2)+O([ut&1�2]3)

as ut&1�2 � 0. From this and from (33) it follows that

Vt0, \(x, y)=

?2i

:3

r, s=1

(1&1\, 0)r, s eiu2

r, s �4tW(ei?�4ur, s �2 - t )

=?2i

:3

r, s=1

(1&1\, 0)r, s[1+H t

r, s(x, y)]

=?2i

:3

r=1

(wF\r (w))w=0+

?2i

:3

r, s=1

(1&1\, 0)r, s H t

r, s(x, y)

=?

2i:3

r, s=1

(1&1\, 0)r, s

iei?�4( |x& yr |+| y& ys | )

2 - t 1 (3�2)

+O([ut&1�2]3)

=- ? ei?�4

2 - t _ :3

r=1

( |x& yr |+| y& yr | )(wF\{(w))w=0&

+O([ut&1�2]3)

=O([ut&1�2]3)

where u=max((x) , ( y) ). For what concerns the computation of the termVt

1, \(x, y) it follows that:

V t1, \(x, y)=

12

:3

r, s=1

(1&1\, 0)$r, s |

R+i0e&ik2teikur, s dk

=- ?

2 - it:3

r, s=1

(1&1\, 0)$r, s eiu2

r, s �4t


For what regards the computation of the remainder term V tR, \(x, y) we

observe that the functions (1�k)(1&1\, k)R

r, s have a zero at k=0 andasymptotically behaves like k&1 as k goes to infinity; from this fact, byintegrating by parts and by means of the stationary phase theorem, asut&1�2 � 0, it follows that V t

R, \(x, y)=O(u�t3�2). Collecting all theseresults, it follows that:

U t\(x, y)=

- ?

- it _iei |x& y| 2�4t&12

:3

r, s=1

(1&1\, 0)$r, s eiu2

r, s �4t&+O([ut&1�2]3)

=- ?

- it _i&12

:3

r, s=1

(1&1\, 0)$r, s+O(u2t&1)&+O([ut&1�2]3)

=O([ut&1�2]3)

since (see formulas (33), (34) and (35))

:3

r, s=1

(1&1\, 0)$r, s=2i(2F1(0)+F2(0))=2i

We underline that the bound (36) is uniform with respect to \ as a resultof the uniform bound (32).

From (36) it follows the uniform bound (4) for a suitable _ largeenough. By applying the same arguments to

e&itH\[H\&(*+i0)]&1 Pc,

=|#

e&izt

z&(*+i0)[H\&z]&1 , dz, * # [d, D]

we obtain the uniform bound (5).

6. SPLITTING VANISHING AND LOCALIZATION

In this section we explicitly compute the solutions of Eq. (11) for thedouble well model discussed in the previous section where, for the sakeof argument, we fix a=1 and b=2; for such values we have that*1=&w2

1 �4a2=0.6351. We fix also the time-dependent perturbation suchthat

g(x)=/(0, a)(x) and f (x)=/(0, a)(x)+$/(&a, 0)(x) (37)


where &1<$<1 is a given parameter, and v(t) is a kicked-type function,for instance

v(t)= :N

n=&N, n{0

cneint, cn=N2n

sin(n?�N)(1&(&1)n), N=10

In order to satisfies Hypothesis H2 we assume + such that n++*1 {0,n=0, \1, \2,..., \N, and \ large enough.

In particular, we consider here two cases: the case of small perturba-tion regime, where the strength = of the periodic part of the perturbationis of the same order of the splitting |; and the case of critical perturbationregime, where = is of the same order of the square root of |. In thefollowing, the strength ' of the static part of the perturbation is assumedto be of the same order of the splitting |.

As we will show, in the first case we observe that the periodic part ofthe perturbation doesn't actually affect the dynamics, that is we observeagain the beating effect (see Theorem 3 and the following remark); in con-trast, in the second case, we obtain the splitting vanishing for a criticalvalue of the parameters (see Theorem 5).

We remark that in the double well model considered in the previoussection and in the large barrier limit \ � +� (corresponding to theDirichlet condition at x=0) we have that *1, 2 t*D=&w2

1 �4a2 and�$x)t�D(\x), where *D is the eigenvalue of H +

D with associatenormalized eigenvector �D . From this fact and from (37) it follows that theelements of the matrices F and G are such that

G+, + tg0 , G+, & tG&, + tG&, & t0

and

F+, + tf0 , F&, & t$f0 , F&, + tF+, & t0

in the large barrier limit, where

f0= g0=(�D , /(0, a) �D) H

In order to compute the terms (9) and (10), let

Pc f�\= f�\&�+(�+ , f�$ H&�&(�& , f�\) H


then it follows that

( f�\ , KmPc f�+) H=( f�\ , Km f�+) H

&( f�\ , Km�+) H (�+ , f�+)H

&( f�\ , Km�&) H (�& , f�+)H

where an explicit computation gives

Km�+=2|

m2+2&|2 �&+2m+

m2+2&|2 �+

Recalling that |t0 and (�\ , f��) H t0 in the large barrier limit, then

( f�+ , KmPc f�+) H t( f�+ , Km f�+)H&2m+

m2+2&|2 |( f�+ , �+) H |2

and

( f�+ , KmPc f�&)Ht( f�+ , Km f�&) H

&2|

m2+2&|2 ( f�& , �&) H (�+ , f�+) H

t( f�+ , Km f�&) H

Therefore, in the large barrier limit

U\, $+)tu\, \(+)= :m{0

|cm | 2 ( f�\ , Km f�$ H (38)

where cm are the Fourier coefficients of v(t). Recalling that V is an evenfunction, then we have that Km=SKmS where (Sf )(x)= f (&x); hence

( f�+ , Km f�+) H=u+, m ( f�& , Km f�&) H=$2u+, m

( f�& , Km f�+) H=$u&, m ( f�+ , Km f�&) H=$u&, m

in the large barrier limit where we set

u+, m=(/(0, a) �D , Km/(0, a) �D) H

u&, m=(/(0, a) �D , KmS(/(0, a) �D)) H


If we set u\=�m{0 |cm |2 u\, m then the matrices M and Mper have theform

Mt|M 0+=2M1, M0=\'g0 �|&1

&10 + , M 1=\ &u+

&$u&

&$u&

&$2u+ +and

Mpert=v(+t) M per, 0+=2Mper, 1(+t), Mper, 0=\ f0

00

$f0+ (39)

and Mper, 1(+t) is a bounded, uniformly with respect to \, = and ', periodicmatrix with mean value zero.

6.1. Small Perturbation Regime

We consider here the small perturbation regime where = is of the sameorder of the splitting |. We underline that, in such a case, by means of aperturbative argument, the assumptions H5 is true.

We have the following result.

Theorem 3. Let J(t)=e&i|M0tB(0), then, in the limit of smallperturbation and large \ such that =�| and '�| have a finite limit, we havethat

|B(t)&J(t)|�C|, \t # [0, T]

for some positive constant C independent of t, =, + and \.

Proof. In order to prove this theorem we simply apply the averagingperturbative method to the Eq. (13), where the mean value of Mper is equalto zero.

Remark. As a result of Theorems 2 and 3 it follows that we observeagain a beating motion with shorter period given by 2?�(- |2+'2g2

0�4).

6.2. Critical Perturbative Regime

We assume here that the strength = of the periodic perturbation is ofthe same order of the square root of the splitting |; in particular weassume the limit of small perturbation and large \ such that

=2�| � k # R+ and '�| � ` # R (40)


In such a case we have that:

Theorem 4. Let J(t)=e&iMtB(0), then, in the limit of small pertur-bation and large \ satisfying (40), we have that

|B(t)&J(t)|�C - |, \t # [0, T] (41)

for some positive constant C independent of t, =, + and \.

Proof. In order to prove this theorem we remark that Eq. (13) takesthe form

{.* =+B4 =[&iM+iM per(.)] B, B(0)=A(0)

(42)

where Mt=2[k&1M0+M 1], k is such that =2�| � k, and Mper(.) is aperiodic matrix with order = and mean value equal to zero. Let

D=B+=K(.) B, K(.)=\K+, +(.)K&, +(.)

K+, &(.)K&, &(.)+ (43)

be a linear transformation where K is a bounded matrix defined below. For= small enough it follows that this transformation is invertible with inverse

B=[1+=K(.)]&1 D=D+=K� (.) D (44)

for some bounded matrix K� . From (42) and from (43) it follows that

D4 =B4 +=K(.) B4 +=+�K�.

B

=_iMper(.)+=+�K�. & B+[&iM+i=K(.) Mper(.)] B&i=K(.) MB

Recalling that M per(.) is of order = and M is of order =2 then the first termof the above equation is of order =, the second one is of order =2 and thethird one is of order =3. By choosing

K\, \(.)=&i

+= |.

0M per

\, \(%) d% (45)


we have that the above equation takes the form

D4 =[&iM+i=K(.) Mper(.)] B&i=K(.) MB

=[&iM+i=K(.) Mper(.)] D+S D

S==[&iM+i=K(.) Mper(.)] K� (.)&i=K(.) M(1+=K� (.))

where the first term is of order =2 and the second one, that is the matrixS, is of order =3. We apply now the averaging perturbative method for anyt # [0, T], where T=2?�| is of order 1�=2, obtaining that

|D(t)&exp[&i(M&=KMper) t] D(0)|�C=2, \t # [0, T] (46)

for some positive constant C. The term KM per denotes the mean value ofthe term K(.) Mper(.); by integrating by parts we have that

KM per=1

2? |2?

0K(.) Mper(.) d.

=&i+=

12? |

2?

0 \|.

0Mper(%) d%+ Mper(.) d.

=&i+=

12? _|

.

0Mper(%) d%&

2?

0

+i

+=1

2? |2?

0Mper(.) \|

.

0Mper(%) d%+ d.

= &MperK

From this fact and since Mper and K are, up to a term of order, respec-tively, =2 and =, diagonal matrices (see Eqs. (39) and (45)) then we havethat

KMper=O(=2) (47)

From (43), (44), (46) and (47) it follows that

|B(t)&e&iMtB(0)|�|B(t)&D(t)|

+|D(t)&e&iMt D(0)|+|e&iMt(D(0)&B(0))|�C=, \t # [0, T]

for some positive constant C.

Remarks. �� We underline that from the above result the validityof assumption H5 follows too.


�� As a consequence of equations (14), (15) and (41) it follows thatthe localization phenomenon and the beating effect depend on the eigen-values l1 and l2 of M. In particular, if |I(l1&l2)|�| is large enough thenwe have a localization result; in contrast, if |I(l1&l2)|<<| then weobserve a beating effect depending on R(l1&l2).

Now, it turns out that when the time dependent perturbation isasymettrical, i.e., ${ \1, then the phenomenon of the vanishing of thesplitting occurs. More precisely:

Theorem 5. For any non-resonant +*>+0 , for some +0>0 andany \* # I large enough then there exist =*==*(+*, \*) and '*='*(+*, \*) such that we have the vanishing of the splitting of the twoeigenvalues of M, i.e., l1=l2 .

Proof. Indeed, recalling that the leading term of M has the form

Mt\ 'g0&=2u+

&|&=2 $u&

&|&=2 $u&

&=2 $2u+ + (48)

in the large barrier limit, then the eigenvalues l1, 2 of M are such thatl1, 2 t

12 |L1, 2 where

L1, 2=z&ku+(1+$2)\- (z&ku+(1&$2))2+4(1+k $u&)2

where '�| � `, z=`g0 and =2�| � k. Hence, we have that L1=L2 when

z*=z(k*)=2i(1+k* $u&)+k*u+(1&$2) (49)

and k* # R+ is such that Iz(k*)=0, i.e.,

k*=2�[2$Ru&+(1&$2) Iu+] (50)

In fact Ru& r0 and Iu+>0 for + not too small (see Fig. 1). By means ofa continuity argument the vanishing of the splitting, i.e., l1=l2 , follows forsome =* and '*.

Remarks. �� In Fig. 2 we fix \=105, in such a case we have that*2=&0.6349, *1=&0.6351 and |=8.5_10&5. For +=+*=0.8 formulas(49) and (50) give that k*=0.408 and z*=3.078. For such values of theparameters we observe the exact crossing (see Fig. 2a).


File: 822J 429326 . By:XX . Date:15:02:01 . Time:07:57 LOP8M. V8.B. Page 01:01Codes: 626 Signs: 233 . Length: 44 pic 2 pts, 186 mm

Fig. 1. Broken and full lines represent, respectively, the functions Ru&(+) and Iu+(+). Itappears that the condition k+>0 (see Eq. (50)) is satisfied provided + is not too small;indeed Ru&(+)r0 and Iu+(+)>0.


File: 822J 429327 . By:XX . Date:08:01:01 . Time:14:55 LOP8M. V8.B. Page 01:01Codes: 698 Signs: 345 . Length: 44 pic 2 pts, 186 mm

Fig. 2. We plot the real (broken line) and imaginary part (full line) of L1, 2 . In figure (a), for+=+*, we have an exact crossing; vanishing of the real part of the splitting and avoidedcrossing of the real part of the eigenvalues are observed, respectively, for +>+* (figure (c))and +<+* (figure (b)).


�� In the case of f symmetric, i.e., $=1, we don't have the vanishingof the splitting because Ru&=0 (see Fig. 1). Because the perturbation termappears at the second order, the same argument applies in the case $=&1too.

�� We remark that, in the very large perturbation regime, i.e., =2>>|(provided that |�=3<<1 in order to have Theorem 2), we have a localizationresult. In particular, by assuming '=0 and $=0 for the sake of argument,then we have l1 t0 and l2 t&=2u+ . From Theorem 2 we have that theleading terms of the solutions A\(t) are given by:

A\(t)tc\, 1e&il1t+c\, 2e&il2t, \t # [0, T]

where

c+, 1=l1 A+(0)&|A&(0)

l1&l2

c+, 2=&l2A+(0)+|A&(0)

l1&l2

c&, 1=&|A+(0)&l2A&(0)

l1&l2

c&, 2=|A+(0)+l1A&(0)

l1&l2

From these facts it follows that, if the state is initially prepared in the leftwell, i.e., A+(0)=0 and A&(0)=1, it stays in the same well for a timelarger than the unperturbed beating period T; that is we generically obtainthe destruction of the beating effect because of localization on one well.

�� We remark that a similar result has been obtained in the largeperturbation limit and by neglecting the term Rc in (8) which couples thediscrete spectrum with the continuous one (see ref. 9 and the referencestherein). In such a way we obtain the simpler two-level model

{iA4 +=&|A&+=v(+t) f0A+

iA4 &=&|A+

(51)

By assuming v(t)=cos(t) (for a recent study where v is a quasi-periodicfunction see ref. 16) the system (51) takes the form :* \=i|:� e\iq(t) wherewe set :+(t)=A+(t) eiq(t), :&(t)=A&(t) and q(t)==f0�+ sin(+t). Byassuming that |<<+ and by means of the mean value approximation, thisequation has approximate solutions of the form :0 cos(|~ t+.0), for some:0 and .0 , for any t # [0, c|&1], for some c, where

|~ =|2? |

&?

?ei(=f0 �+) sin({) d{=|J0 \=f0

+ +


J0 is the zero-th Bessel function. It appears that for a two-level periodicallydriven model the beating period is given by Tper=TJ &1

0 (=f0 �+) whereT=2?�| is the beating period of the unperturbed double-well model. Inthe large perturbation limit such that =�+ � � then Tper tT - (=f0 �+)>>T, in such a case we have the interruption of the beating effect; weremark that the interruption of the beating effect appears also for inter-mediate values of =, such that =f0�+ coincides with a zero of the zero-thBessel function.

ACKNOWLEDGMENTS

This work is partially supported by the Italian MURST and INDAM-GNFM. V. Grecchi is supported by the INFN and by the University ofBologna (funds for selected research topics); A. Sacchetti is supported bythe program ``Progetti di ricerca avanzata o applicata'' of the University ofModena and Reggio Emilia.

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